Friday Puzzle -- A special magic square

Clue 1

Alex Y

The special property of the magic square in question involves another magic square.

Re: Friday Puzzle -- A special magic square

Bill Earl

Would one need a digital computer to solve this?

Re: Friday Puzzle -- A special magic square

Alex Y

Would one need a digital computer to solve this?

I don't think so, but maybe I'm missing the implication. What did you have in mind?

Re: Friday Puzzle -- A special magic square

Bill Earl

The property I am thinking of has to do with computations on digits (as opposed to numbers)

You may have found another property

Alex Y

Not the one I had in mind, but explain yours.

Re: You may have found another property

Bill Earl

The sum of the digits for each row, column and diagonal add up to 18.

Re: Friday Puzzle -- A special magic square

David Weaver

Haven't tried to do anything with it yet, but interesting by inspection that every single row, column and diagonal is the same sum, making the sum of the outside of the box sides all identical.

Maybe that's a property of these in general, though.

Re: Friday Puzzle -- A special magic square

David Weaver

Just read bill's last comment, only an hour late after he already said that!

Re: You may have found another property

Alex Y

I think that having a constant sum of digits is not very rare for magic squares -- basically, it says you have the same number of "carries" in each sum. For instance,

shares that property, with the same sum of digits. The property I am looking for works only, I believe, for the given magic square and its seven reflections and rotations, at least among 3x3 squares with two or fewer digits.

Re: Friday Puzzle -- A special magic square

Larry Barrett

It is interesting that only 4 digits are used: 1, 2, 5, 8.

And the numbers in the row and column associated with each corner use the same set of digits. For example, the numbers associated with the top left cell (25) are (8, 12) in the top row and (2, 18) in the left column. Same for all 4 corner cells, same for the center cell (15) and the remaining numbers in the center column and row (8, 22), (2, 28).

Clue 2

Alex Y

The other magic square referred to in clue 1 is

As a reminder, the magic square in the original puzzle is:

Re: Clue 2

Larry Barrett

Subtract the second magic square from the first magic square, cell by cell.

Result is:

15 3 6

-1 8 17

10 13 1

A third magic square! and the sum of numbers (row, column, diag) for this square = 24, which is also equal to the difference between the other two squares: 45 - 21 = 24.

Re: Clue 2

Larry Barrett

Second thoughts - nevermind. Nothing remarkable about adding, subtracting magic squares.

Alex Y

Yeah, hard to imagine a pair of magic squares for which that doesn't work!

But stick with the relationship between these two magic squares. BTW, it is a "one-way" relationship.

Final clue

Alex Y

This only works for English speakers

Answer

Alex Y

In this special magic square, if you write out every number in English, the letter counts of the words also form a magic square.

I speculated that this square was unique, except for rotations and reflections, but then limited that speculation to magic squares with two or fewer digits in each number. For instance, adding 300 to every member of this MS will lead to one in which "three hundred and" is appended to every English spelling of the numbers, adding a constant 15 to the word counts, leading to a MS.